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Control of 2D scalar conservation laws in the presence of shocks


Authors: Rodrigo Lecaros and Enrique Zuazua
Journal: Math. Comp. 85 (2016), 1183-1224
MSC (2010): Primary 35L67, 49J20, 90C31, 49M30, 35L65
DOI: https://doi.org/10.1090/mcom/3015
Published electronically: August 25, 2015
MathSciNet review: 3454362
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Abstract: We analyze a model optimal control problem for a 2D scalar conservation law--the so-called inverse design problem--with the goal being to identify the initial datum leading to a given final time configuration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and finite differences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The efficiency of the method is illustrated by numerical experiments.


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Additional Information

Rodrigo Lecaros
Affiliation: BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E-48009, Bilbao, Basque Country, Spain — and — CMM - Centro de Modelamiento Matemático. Universidad de Chile (UMI CNRS 2807), Avenida Blanco Encalada 2120, Casilla 170-3, Correo 3, Santiago, Chile
Email: rlecaros@dim.uchile.cl

Enrique Zuazua
Affiliation: BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E-48009, Bilbao, Basque Country, Spain — and — Ikerbasque - Basque Foundation for Science, Maria Diaz de Haro, 3. 48013 Bilbao, Basque Country, Spain
Email: zuazua@bcamath.org

DOI: https://doi.org/10.1090/mcom/3015
Received by editor(s): April 30, 2014
Received by editor(s) in revised form: October 3, 2014, and November 15, 2014
Published electronically: August 25, 2015
Additional Notes: The first author was partially supported by Basal-CMM project, PFB 03
This work was done while the second author was visiting the CIMI (Centre International de Mathématiques et Informatique) of Toulouse (France) and the University of Erlangen-Nürnberg within the Humboldt Research Award program
This work was supported by the Advanced Grants NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, PI2010-04 and the BERC 2014-2017 program of the Basque Government, the MTM2011-29360-C02-00 and SEV-2013-0323 Grants of the MINECO
Article copyright: © Copyright 2015 American Mathematical Society

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